VOLUME 76, NUMBER 21
PHYSICAL REVIEW LETTERS
20 MAY 1996
Universality of Flux Creep in Superconductors with Arbitrary Shape and Current-Voltage Law
Ernst Helmut Brandt
Max Planck Institute für Metallforschung, Institut für Physik, D-70506 Stuttgart, Germany
(Received 12 January 1996)
The nonlinear and nonlocal diffusion equation for the relaxing current density Jsr, td in long
superconductors of arbitrary cross section in a constant perpendicular magnetic field Ba is solved
exactly by separation of variables in the electric field Esr, td ­ fsrdgstd. This solution includes the
limiting cases of longitudinal and transverse geometries and applies to the current-voltage laws E ~ J n
ranging from Ohmic (n ­ 1) to Bean-like (n ! `) behavior. The electric field profile fsrd weakly
depends on n and becomes universal for n exceeding ø5. At large times t one finds E ~ 1yt nysn21d
and J ~ 1yt 1ysn21d for n . 1, and E ~ J ~ exps2tyt0 d for n ­ 1. The contour lines of the creeping
Esr, td coincide with the field lines of Bsr, td in the remanent state Ba ­ 0. [S0031-9007(96)00286-4]
PACS numbers: 74.60.Ge, 74.25.Ha
Type-II superconductors, in particular the high-Tc superconductors, in general are not ideal conductors of electric current when put into a magnetic field Ba . Nonzero
resistivity appears when the Abrikosov vortices induced
by Ba depin due to a current density J exceeding a critical value Jc , or by thermal activation occurring also
at J , Jc . The resulting motion of vortices generates
an electric field E ­ By, where B is the flux density
and y the drift velocity. A superconductor may thus
be characterized by its current-voltage law EsJd, which
in general is nonlinear and strongly depends on Ba and
on the temperature T . This EsJd characteristic is most
sensitively extracted from magnetization measurements,
e.g., by slowly increasing or decreasing Ba std and then
keeping it at a constant (or zero) value at times t $
0. During such creep experiments, the magnetic moment caused by persistent currents relaxes with an approximately logarithmic time law when EsJd is strongly
nonlinear, and exponentially when EsJd ­ rJ is linear
(Ohmic).
Recently, Gurevich and co-workers have shown that
in both longitudinal [1,2] and transverse [2,3] geometries, corresponding to zero and unity demagnetizing factors, respectively, the relaxing electric field Esr, td is
universal if EsJd is sufficiently nonlinear. This means
that after some transient time t1 , Esr, td ­ fsrdgstd separates into a universal profile fsrd and a time dependence
gstd ~ 1yst1 1 td ø 1yt. The relaxingRcurrent density
1
Jsr, td and magnetic moment mstd ­ 2 r 3 Jsr, tdd 3 r
are then obtained by inserting this general electric field
into the material law J ­ JsEd. This separation was
found to be exact if one has ≠Ey≠J ­ EyJ1 with constant J1 , corresponding to an exponential law EsJd ­
Ec expfsJ 2 Jc dyJ1 g, and if E depends only on one spatial coordinate. This applies to slabs [1,2] or cylinders
[4] in parallel field, and to thin strips or circular disks in
perpendicular field [2,3]. The same universality of creep,
and the existence in transverse geometry of “neutral lines”
along which the flux-density Bsr, td stays constant during
creep, was recently demonstrated by magneto-optic obser4030
0031-9007y96y76(21)y4030(4)$10.00
vation and by computation to occur also in thin specimens
of square shape [5].
In the present Letter I show that the exact solution of
the creep problem by separation of variables of Esr, td
is possible for a much more general geometry including
specimens of finite thickness and arbitrary cross section
in a perpendicular field, and for all current-voltage laws
EsJd ~ J n , or, more precisely,
(1)
EsJd ­ Ec jJyJc jn sgnJ ,
yielding JsEd ­ Jc jEyEc j1yn sgnE. This power law is
observed in numerous experiments and was considered
elsewhere in connection with creep [2,6] and flux penetration and ac susceptibility [4,7–10]. It corresponds
to a logarithmic current dependence of the activation
energy UsJd ­ Uc lnsJc yJd, which inserted into an Arrhenius law yields EsJd ­ Ec exps2UykT d ­ Ec sJyJc dn
with n ­ Uc ykT ¿ 1; it contains only two parameters
Ec yJcn and n; and it interpolates from Ohmic behavior
(n ­ 1) over typical creep behavior (n ­ 10, . . . , 20) to
“hard” superconductors with Bean-like behavior (n ! `).
A power law avoids the unphysical behavior at J ­ 0 exhibited by the exponential law originally used in Refs. [1–
3] to achieve separation of variables in the creep problem.
Consider the rather general geometry of a long (along
z) superconductor of arbitrary cross section S in a perpendicular field Ba directed along y, e.g., a rectangular bar
filling the volume jxj # a, j yj # b, jzj , `, cf. Fig. 1.
The current density Jsx, yd, electric field Esx, yd, and vector potential Asx, yd then point along z and the magnetic
induction Bsx, yd ­ = 3 A is in the x-y plane. Throughout this Letter I shall assume B ­ m0 H, i.e., disregard
the finite value of the lower critical field Bc1 . This approximation is excellent when everywhere inside the superconductor one has B . 2Bc1 . From J ­ = 3 H ­
m21
0 = 3 s= 3 Ad one then gets for our geometry J ­
2
2m21
0 = A, which has the solution
Z
d 2 r 0 Qsr, r 0 dJsr 0 d 2 xBa ,
Asrd ­ 2m0
S
Qsr, r 0 d ­ s1y2pd lnjr 2 r 0 j .
© 1996 The American Physical Society
(2)
VOLUME 76, NUMBER 21
PHYSICAL REVIEW LETTERS
20 MAY 1996
µ
t
Esr, td ­ Ec fn srd
t1 1 t
FIG. 1. The field lines of the induction B during penetration
of perpendicular flux into long bars with side ratios bya ­ 2
(left) and bya ­ 0.4 (right) in the Bean limit (n ­ 51) for
applied fields Ba ­ 0.3, 0.7 (Ba ­ 0.05, 0.1, 0.2, 0.4, in units
m0 Jc a. The field of full penetration is Bp ­ 0.96 (0.87, 0.72,
0.49, 0.33) for bya ­ 4 (2, 1, 0.4, 0.2). The dashed rectangle
marks the specimen boundary. The bold line (composed of
the merging contour lines jJjyJc ­ 0.5, . . . , 0.3) shows the
penetrating flux front inside which J ­ 0 and B ­ 0.
Here and in the following r denotes sx, yd and the
integration is over the specimen cross section S. The
Ù ­ ≠By≠t ­ 2= 3 E in this geometry
induction law B
reduces to AÙ ­ 2E. Combining this with Eq. (2) and
with the material law E ­ EsJd or J ­ JsEd, one obtains
the equation of motion for Esr, td,
Z
≠
Ù a . (3)
d 2 r 0 Qsr, r 0 d JfEsr 0 , tdg 1 x B
Esr, td ­ m0
≠t
S
Figure 1 shows the magnetic field lines in and around
a rectangular superconductor during flux penetration obtained from a similar integral equation for A.
Ù a ­ 0,
In creep experiments one has Ba ­ const; thus B
and Eq. (3) simplifies to
Z
Ù 0 , td ≠J .
Esr, td ­ m0 d 2 r 0 Qsr, r 0 dEsr
(4)
≠E
This integral equation describes the nonlinear and nonlocal diffusion of Esr, td during flux creep. For the
power law EsJd (1), one has ≠Jy≠E ­ 1ys≠Ey≠Jd ­
sJc yhEd sEyEc d1yh , yielding
m 0 Jc Z 2 0
Ù 0 , tdE s1ynd21 . (4a)
d r Qsr, r 0 dEsr
Esr, td ­
1yn
nEc
Remarkably, this nonlinear integral equation can be
solved exactly by the ansatz E ­ fsrdgstd, yielding
∂nysn21d
.
(5)
If we choose t ­ m0 Jc Sy4psn 2 1dEc we obtain fh srd
from
4p Z 2 0
d r Qsr, r 0 dfn sr 0 d1yn .
(6)
fn srd ­ 2
S
This nonlinear integral equation is easily
by iterR solved
sm11d
smd1yn
­ 2s4pySd Qfn
starting
ating the relation fn
s0d
s1d
s2d
with fn ­ 1. The resulting series fn , fn , . . . , converges rapidly if n . 1. For m ¿ 1 one has approxism11d
smd
srd ø fn srd s1 1 cynm d with c ø 1. For
mately fn
n ¿ 1 (practically for n $ 5) the shape fn srd of the electric field becomes a universal function which depends
only on the specimen shape but not on the exponent n,
2Z 2 0
fn$5 srd ø f` srd ­ 2
d r lnjr 2 r 0 j .
(7)
S S
In the Ohmic case n ­ 1, the ansatz (5) makes no
sense since the exponent nysn 2 1d diverges. For this
special case the integral Eq. (4) is linear since the factor
≠Jy≠E ­ s ­ 1yr is the constant Ohmic conductivity.
This linear integral equation is solved by the ansatz
Esr, td ­ E0 f0 srd exps2tyt0 d ,
(8)
yielding the relaxation time t0 ­ m0 sSyL, where L and
f0 srd are the lowest eigenvalue and eigenfunction of the
linear integral equation
2L Z 2 0
f0 srd ­ 2
d r lnjr 2 r 0 jf0 sr 0 d .
(9)
S S
From (1) and (5) the current density becomes
∂1ysn21d
µ
t
1yn
sgnfn srd .
Jsr, td ­ Jc j fn srdj
t1 1 t
(10)
For n ¿ 1, t ¿ t1 the time factor in (10) equals 1 2
1
n21 lnstytd and one has jJj ø Jc everywhere.
The magnetization along y per unit length along z is
Z
Mstd ­ ŷmstdyLz ­
xJsx, y, td dx dy .
(11)
S
1
1R
In (11) the prefactor 2 of mstd ­ 2 r 3 Jsr, tdd 3 r was
compensated by the contribution to m of the U-turning
currents at the far away ends of the bar at z ­ 6Lz y2 ¿
a, b. The resulting factor of 2 in M was sometimes
missed in previous work on slabs.
These results apply to arbitrary specimen cross section
S. For the realistic rectangular cross section 2a # x #
a, 2b # y # b, S ­ 4ab, with Ba along y, the symmetry Jsx, yd ­ 2Js2x, yd ­ Jsx, 2yd (and same symmetry for E and A) allows one to restrict the integration to
the quarter 0 # x # a, 0 # y # b by using the symmetric kernel
Qsym sr, r 0 d ­
2 d sx 2 1 y 2 d
sx 2 1 y2
1
2
1
,
ln 2
2
2 d sx 2 1 y 2 d
4p sx1
1 y2
1
1
(12)
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VOLUME 76, NUMBER 21
PHYSICAL REVIEW LETTERS
20 MAY 1996
with x6 ­ x 6 x 0 , y6 ­ y 6 y 0 . One now has in (5)
(13)
t ­ m0 Jc abyfpsn 2 1dEc g ,
Z a dx 0 Z b dy 0 sx 2 1 y 2 d sx 2 1 y 2 d
2
1
1
ln 1
fn sx, yd ­
2 1 y 2 d sx 2 1 y 2 d
a 0 b
sx2
0
1
2
2
(14)
3 fn sx 0 , y 0 d1yn .
Before discussing the rectangular cross section I first
show that in the limits of longitudinal (b ¿ a) and transverse (b ø a) geometry the above expressions reduce to
known and new results. In both limits, the functions J,
E, A, Bx , and By depend only on x. One may thus put
y ­ 0 writing,
Rae.g., Esx, yd ø Esx, 0d ­ Esxd, and one
gets M ­ 4b 0 xJsxd dx.
In the transverse limit (strip with thickness 2b ø
a), the reduced one-dimensional kernel is Qtrans sx, x 0 d ­
2bQsym sx, 0, x 0 , 0d ­ sbypd lnfjx 2 x 0 jysx 1 x 0 dg. This
kernel was used in calculations of creep [3,10], flux
penetration [9], and nonlinear [10] and linear [11,12] ac
response of superconducting strips. In the creep solution
(5) one now has t (13) and the shape fn sxd is determined
by
Ç
Ç
1 Z a 0
x 1 x0
fn sxd ­
dx ln
(15)
fn sx 0 d1yn .
a 0
x 2 x0
For n ¿ 1 this reproduces the E of Ref. [3],
µ
∂
a1x
x a2 2 x 2
m0 Jc ab 1
ln
1
ln
.
Esx, td ­
psn 2 1d t
a2x
a
x2
(16)
In the longitudinal limit (slab Rwith b ¿ a), the reduced
`
kernel is Qlong sx, x 0 d ­ 2 0 dy 0 Qsym sx, 0, x 0 , y 0 d ­
1
0
0
0
(the
min2 sjx 2 x j 2 jx 1 x jd ­ 2Minsx, x d
0
imum value of x, x ), which has the property
00
­ dsx 2 x 0 d. One now has in (5) t from
Qlong
(13) and fn sxd from
p Z a 0
fn sxd ­
dx Minsx, x 0 dfn sx 0 d1yn ,
(17)
ab 0
0 # x # a, fs2xd ­ 2fsxd. For large exponents n ¿
1 this reproduces the result of Gurevich [1,2],
m0 Jc xs2a 2 jxjd
Esx, td ­
.
(18)
n21
2t
The unphysical dependence of t (13) and fn sxd (17),
but not of Esx, td (18), on the specimen length 2b may
be removed by redefining t and fn , multiplying the
integrals in (14) and (17) by a factor 1 1 pby4a, and
dividing t (13) by the same factor. This definition leaves
the transverse limit unchanged but replaces the prefactor
pyab in (17) by p 2 y4a2 . The integral equation (17)
is equivalent to a differential equation with boundary
conditions,
fn s0d ­ fn0 sad ­ 0 , (19)
fn00 sxd ­ 2k 2 fn sxd1yn ,
where k ­ py2a in our new definition of t. The
solutions of (19) in the Ohmic (n ­ 1) and Bean (n ­ `)
limits look very similar, f1 sxd ­ const 3 sinspxy2ad
and f` sxd ­ sp 2 y8a2 dxs2a 2 jxjd, cf. Fig. 2.
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FIG. 2. Profiles of the electric field E ~ fn sxd and current
density J ~ fn sxd1yn during creep in longitudinal (17) and
transverse (15) geometries for exponents n ­ 1, 2, 5, and `.
Creep profiles fn sxd obtained from (14) and (17)
are depicted in Fig. 2 for exponents n ­ 1, 2, 5, and `.
Notice that the fn sxd only weakly depend on n but are
qualitatively different for the two geometries: In the
longitudinal limit the fn sxd are maximum at the boundary
x ­ a, whereas the transverse fn sxd exhibit a maximum
inside the specimen,
e.g., at x ­ 0.735a for n ­ 1 [11]
p
and x ­ ay 2 ­ 0.707a for n ! ` [3].
The contour lines of the creeping Esx, y, td inside
and outside rectangular bars with aspect ratios bya ­
2, 1, and 0.2 are depicted in Fig. 3 for the Bean limit
n ¿ 1. These contour lines of E are also the field lines
of the induction Bsx, y, td in the remanent state. Namely,
for n ¿ 1 one gets from (2) and (12) the vector potential
Z aZ b
Qsym sr, r 0 ddy 0 dx 0 2 xBa ,
Asx, yd ø 2m0 Jc
0
0
(20)
because one has J ø Jc sgnx during creep. The lines
A ­ const are the field lines of B 5 = 3 sẑAd. For
Ba ­ 0 these lines coincide with the contour lines of E ­
Ù since during creep the time dependence separates,
2A,
Esr, td ­ fsrdgstd, Eq. (5). For nonzero applied field Ba ,
the lines E ­ const are still as depicted in Fig. 3. The
field lines of B for Ba near the penetration field Bp look
VOLUME 76, NUMBER 21
PHYSICAL REVIEW LETTERS
20 MAY 1996
and induction B ' z, during flux creep is presented
for nonlinear conductors with constitutive laws E ~ J n
and B ­ m0 H and with rectangular cross section 2a 3
2b in a perpedicular field Ba k y. The obtained twodimensional universal profiles of E during creep (Fig. 3)
explain how the one-dimensional profiles Esx, td of the
two limiting geometries b ¿ a and b ø a (Fig. 2)
come about. In particular, the maximum of Esx, td in
transverse geometry, corresponding to a maximum in
the nearly constant J ­ Jc sEyEc d1yn ø Jc and in the
energy dissipation EJ ~ E 111yn ~ J n11 , is related to the
maximum of the vector potential A k z (2) caused by
this current and depicted in Fig. 3. This theory is easily
extended to circular disks or cylinders with finite height
by modifying the kernel Qsr, r 0 d.
The origin of the well known sign reversal of the
perpendicular component By at the surfaces y ­ 6b near
the edges x ­ 6a, occurring in the remanent state and
leading to a neutral line on which B ­ Ba stays constant
during creep [3,5], becomes clear from the magnetic field
lines in Fig. 3, which coincide with the contour lines of
E and A during creep. Interesting further effects [13] are
expected at positions with small B if a more general BsHd
with finite Bc1 is accounted for. Work in this direction is
under way.
FIG. 3. The contour lines of the electric field E ­ 2AÙ
during creep in long rectangular superconductors with side
ratios bya ­ 2, 1, and 0.2 (from top) for n ­ 51 and arbitrary
constant applied field Ba . These lines coincide with the
magnetic field lines in the fully saturated remanent state Ba ­
0. Bottom: 3D plot of E (and of A) for bya ­ 0.2.
similar as depicted in Fig. 1, bottom row; they are nearly
parallel lines slightly inflated inside the specimen.
All these analytical results are confirmed by computations of creep from Eq. (4): During constant ramp rate
BÙ a fi 0 one has E ­ x BÙ a in the fully penetrated state.
When Ba is held constant, the straight E profile starts to
curve down near the edge and, after a short transition time
t1 , it reaches the universal profiles depicted in Figs. 2 and
3. During creep the shapes of E, J, and B practically do
not change over many decades of the creep time t, but
their amplitudes decrease according to Eqs. (5) and (10).
In particular, for n ¿ 1 and t ¿ t1 one has E ~ 1yt and
1
M ~ J ~ stytd1ysn21d ø 1 2 n21 lnstytd.
In conclusion, the exact solution for the electric field
Esx, y, td k z, and thus for the current density J k z
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